Problem 71 - College Algebra and Calculus: An Applied Approach

Question

Refrigeration When food is placed in a refrigerator, the time required for the food to cool depends on the amount of food, the air circulation in the refrigerator, the original temperature of the food, and the temperature of the refrigerator. One model for the temperature of food that starts at \(75^{\circ} \mathrm{F}\) and is placed in a \(40^{\circ} \mathrm{F}\) refrigerator is \(T=10\left(\frac{4 t^{2}+16 t+75}{t^{2}+4 t+10}\right), \quad t \geq 0\) where \(T\) is the temperature (in degrees Fahrenheit) and \(t\) is the time (in hours). Sketch a bar graph showing the temperature of the food when \(t=0,1,2,3,4\), and 5 hours. According to the model, will the food reach a temperature of \(40^{\circ} \mathrm{F}\) after 6 hours?

Step-by-Step Solution

Answer

After solving for the temperatures at the different timepoints, we plot them on a bar graph. When we calculate the temperature at t = 6, we find out whether the food reaches the temperature of the refrigerator or not.

Step 1: Calculate Temperature at Different Times

We are given 6 different times \(t=0,1,2,3,4,5\). Substitute these values in the given formula to calculate the temperature at each time. Use the formula: \(T=10\left(\frac{4 t^{2}+16 t+75}{t^{2}+4 t+10}\right)\).

Step 2: Sketch a Bar Graph for Temperatures

Using the temperatures calculated in Step 1, plot the bar graph. On the X-axis, list the time in hours and on the Y-axis list the corresponding temperatures.

Step 3: Calculate the Temperature after 6 hours

We are asked to find if the temperature of the food reaches 40°F after 6 hours. Substitute \(t=6\) into the given equation to find if the temperature reaches 40°F.

Key Concepts

Refrigeration Temperature Model

The refrigeration temperature model is crucial for predicting how quickly food cools when placed in a refrigerator. This particular model is described by the equation \(T=10\left(\frac{4 t^{2}+16 t+75}{t^{2}+4 t+10}\right)\), where \(T\) represents temperature in degrees Fahrenheit, while \(t\) indicates time in hours.
The goal is to determine how the temperature of food changes over time, starting from an initial temperature of \(75^{\circ} \mathrm{F}\), when it's placed in a \(40^{\circ} \mathrm{F}\) refrigerator.

When using this model, consider:

• Initial temperature of the food, which greatly influences how temperature changes happen.
• Refrigerator's consistent temperature, which acts as a limiting factor for how low the food's temperature can ultimately reach.
• Time variable \(t\), that modifies the expression's structure and results in temperature changes.

This model thus reflects an exponential decay pattern, where temperatures decrease over time, influenced by the mathematical structure of the formula.

Algebraic Functions

Algebraic functions link inputs to outputs in a predictable manner using combinations of operations such as addition, multiplication, and exponents.

In this exercise, the function \(T=10\left(\frac{4 t^{2}+16 t+75}{t^{2}+4 t+10}\right)\) is a rational function.
Here, both the numerator and denominator are quadratic polynomials in terms of \(t\). Quadratic polynomials have the general form \(at^2 + bt + c\), where \(a\), \(b\), and \(c\) are constants:

• Numerator: \(4t^2 + 16t + 75\)
• Denominator: \(t^2 + 4t + 10\)

To calculate specific values for \(T\), substitute the desired \(t\) values into the equation. The operations inside the function govern how \(T\) varies, giving insight into how variables interact within a function.

This model exemplifies how algebraic functions can describe real-world phenomena like temperature changes over time.

Graphical Representation

Visualizing the results of calculations aids significantly in understanding data and relationships. In this case, plotting the temperature against time in a bar graph serves as a compelling visual representation of the refrigeration model.

The correct steps for constructing this graph are:

• Identify the time points and corresponding temperatures from calculated data.
• Label the X-axis with time (in hours) and the Y-axis with temperature (in degrees Fahrenheit).
• Draw bars for each time point, ensuring the bar height matches the calculated temperature value.

This bar graph not only illustrates exponential decay but also provides an intuitive visual aid, showing how temperature decreases over time.

Checking the temperature after 6 hours through the graph or calculations gives quick insights into whether the food reaches the refrigerator's limit temperature. By observing the graph, students can visually verify how close the final temperature comes to \(40^{\circ} \mathrm{F}\), reinforcing understanding of both the mathematical model and the real-world implications of fitted functions and their graphs.